How do you study Jazz?

I am starting to learn to play more jazz tunes. How do you guys go about learning jazz songs.

I have a copy of some fakebooks that are great. I want to learn Love for Sale by Cole Porter. I have been listening to the Charlie Parker version for a few months on my MP3 player and decided to learn this song. I was taking lessons when I had a job and my teacher taught me to learn the melody first, then work on the arpeggios of the chords and then to practice walking over the chords. I am curious how others learn to play jazz songs.

As far as walking bass lines for jazz, do you constantly change your lines each time you play the song or do you stick with one version and play it that way each time....I guess that is not improvisation then.

I have a teacher when I can afford one, but my question was more along the lines of the thought process, when playing jazz tunes. Is it better to go in having memorized the lines you will play for a song or is it better to strive for spontaneity derived from a strong foundation in theory and fretboard knowledge? My goal is to be able to walk on a tune and not to play it the same way everytime. Is this a proper approach to playing jazz. If I play it the same way someone else played it, that seems to defeat the purpose of not having everything written out like in alot of pop tunes.

I actually just started taking lessons after playing for about 10 years and building walking lines is the first thing we're starting in on.

My situation is kindof weird, I actually have 2 degrees in music (B.S. and M.M but neither in bass, I don't play upright -yet-) so I have classical theory chops coming out of my ears. However, I'd never spent the time to apply that knowledge to the practical application on the instrument. So what I'm after from my teacher is kindof what you're describing, a command of the theory in such a way that I can improvise the lines with only the changes and tune provided.

I had my first lesson yesterday and it was excellent. I'm looking forward to really digging into this stuff. I've always had this almost deified opinion of jazzers because of kindof command of theory they have to have. Now I get to crack that nut and play the stuff myself.